188 ANNUAL EEPOKT SMITHSONIAN INSTITUTION, 1912. 



always appealed powerfully to dame mathematics; and a keen 

 curiosity, fanned into an intense flame by little bits of apparently 

 incoberent information, bas inspired some of tbe most arduous and 

 prolonged researches. Incentives of this kind have led to the mathe- 

 matics of the invisible, relating to refuiements which are essentially 

 foreign to counting and measuring. The first important refinement 

 of this type relates to the concept of the iiTational, mtroduced by the 

 ancient Greeks. As an instance of a comparatively recent develop- 

 ment along this line we may mention the work based upon Dedekind's 

 defuiition of an infinite aggregate as one in which a part is similar or 

 equivalent to the whole.^ 



Mathematics is commonly divided into two parts called pure and 

 applied, respectively. It should be observed that there are various 

 degrees of purity, and it is very difficult to say where mathematics 

 becomes sufficiently impure to be called applied. The engineer or 

 the physicist may reduce his problem to a differential equation, the 

 student of differential equations may reduce his troubles to a question 

 of function theory or geometry, and the workers in the latter fields 

 find that many of their difficulties reduce themselves to Cjuestions in 

 number theory ^ or in higher algebra. Just as the student of applied 

 mathematics can not have too thorough a training in the pure mathe- 

 matics upon which the applications are based so the student of some 

 parts of the so-called pure mathematics can not get too thorough a 

 training in the basic subjects of this field. 



As mathematics is such an old science and as there is such a close 

 relation between various fields, it might be supposed that fields of 

 research would lie in remote and almost inaccessible parts of this 

 subject. It must be confessed that this view is not without some 

 foundation, but these are days of rapid transjiortation and the 

 student starts early on his mathematical journey. The question as 

 regards the extent of explored country wliich should be studied 

 before entering unexplored regions is a very perplexing one. A life- 

 time would not suffice to become acquainted with all the known fields, 

 and there are those who are so much attracted by the explored regions 

 that they do not find time or courage to enter into the unknown. 



In 1840 C. G. J. Jacobi used an illustration, in a letter ^ to his 

 brother, which may serve to emphasize an important pomt. He 

 states that at various times he had tried to persuade a young man to 

 begin research in mathematics, but this young man always excused 

 himself on the ground that he did not yet know enough. In answer 

 to this statement Jacobi asked this man the following question: 

 "Suppose your family would wish you to marry would you then also 



i"Encyclopc'die des sciences mathematiques," vol. i., pt. 1, 1904, p. 2. 



2"Dcr Urquell aller Mathematik siad die ganzen Zahlcn," Minkowski, Dioptaantische Approximation, 

 1907, pro face, 

 s " Briolwoohsel zwischon C. G. J. Jacobi und M. n. Jacobi," 1907, p. 64. 



